We don’t yet have a complete and consistent theory that combines
quantummechanicsandgravity.However,wearefairlycertainofsome
features that such a unified theory should have. One is that it should
incorporateFeynman’sproposaltoformulatequantumtheoryintermsof
a sum over histories. In this approach, a particle does not have just a
singlehistory,asitwouldinaclassicaltheory.Instead,itissupposedto
followeverypossiblepathinspace-time,andwitheachofthesehistories
thereareassociatedacoupleofnumbers,onerepresentingthesizeofa
waveandtheotherrepresentingitspositioninthecycle(itsphase).The
probabilitythattheparticle,say,passesthroughsomeparticularpointis
foundbyaddingupthewavesassociatedwitheverypossiblehistorythat
passes through that point. When one actually tries to perform these
sums,however,onerunsintoseveretechnicalproblems.Theonlyway
aroundtheseisthefollowingpeculiarprescription:onemustaddupthe
wavesforparticlehistoriesthatarenotinthe“real”timethatyouandI
experience but take place in what is called imaginary time. Imaginary
time may sound like science fiction but it is in fact a well-defined
mathematicalconcept.Ifwetake anyordinary(or“real”) numberand
multiply it by itself, the result is a positive number. (For example, 2
times2is4,butsois−2times−2.)Thereare,however,specialnumbers
(calledimaginarynumbers)thatgivenegativenumberswhenmultiplied
bythemselves.(Theonecalledi,whenmultipliedbyitself,gives−1,2i
multipliedbyitselfgives−4,andsoon.)
One can picture real and imaginary numbers in the following way:
Therealnumberscanberepresentedbyalinegoingfromlefttoright,
withzerointhemiddle,negativenumberslike−1,−2,etc.ontheleft,
andpositivenumbers, 1,2, etc. onthe right. Thenimaginarynumbers
are represented by a line going up and down the page, with i, 2i, etc.
abovethemiddle,and−i,−2i,etc.below.Thusimaginarynumbersare
inasensenumbersatrightanglestoordinaryrealnumbers.
ToavoidthetechnicaldifficultieswithFeynman’ssumoverhistories,
one must use imaginary time. That is to say, for the purposes of the
calculation one must measure time using imaginary numbers, rather
than real ones. This has an interesting effect on space-time: the
distinctionbetweentimeandspacedisappearscompletely.Aspace-time
inwhicheventshaveimaginaryvaluesofthetimecoordinateissaidto
beEuclidean,aftertheancientGreekEuclid,whofoundedthestudyof